New empirical findings in the biodemography of longevity have challenged the traditional models of mathematical demography on which the evolutionary theory of senescence and the demographic modeling of hazard functions have relied. This project seeks to deploy new advances in stochastic process theory, and where necessary extend them, to develop formal demographic models capable of accommodating patterns uncovered by biodemographic research. Aim 1 is to construct stochastic demographic models for the dynamic generation and modification of hazard functions over evolutionary time which can successfully produce hazard functions that taper at extreme ages consistent with biodemographic observations. Aim 2 is to derive individual-level consequences of aggregate hazard function models that generate heterogeneity implicitly and compare their testable consequences. Aim 3 is to integrate models for resource constraints mediated through intergenerational transfers into mathematical formulations of evolutionary theories of senescence and revise and extend them. With an emphasis on testable consequences, this mathematical work seeks to synthesize findings from the other projects of the Program Project and prepare the way for effective use of future human data from sample surveys combining genetic and biological indicators with socio-demographic variables.